It is well known that the use of multiple antennas at the transmitter and/or receiver may significantly boost the performance of a wireless system. Such multiple-input-multiple-output (MIMO) antenna configurations have the potential of both improving data rates and increasing diversity. Precoding is one multi-antenna technique for improving the performance of a MIMO system by transforming the information carrying transmit vector so that the vector better fits the channel conditions. This may be done based on channel information or completely without channel information or some combination thereof. Often, precoding is implemented as performing a linear transformation on the information carrying vector prior to transmission. Such linear transformation is usually represented by a matrix. Precoding is an integral part of LTE (3GPP Long Term Evolution) as well as of WCDMA (Wideband Code Division Multiple Access) and WiMax (Worldwide Interoperability for Microwave Access).
There are two basic types of precoding: codebook based and non-codebook based. Codebook based precoding involves the precoding matrix implementing the linear transformation being selected from a countable and typically finite set of candidate matrices. The set of candidate matrices constitutes the codebook. On the other hand, non-codebook based precoding does not involve any quantization. The precoding element may thus for example be a continuous function of the channel matrix.
Beamforming is a special case of the more general notion of precoding and involves a single information carrying symbol stream being multiplied by a channel dependent vector that adjusts the phase of the signal on each transmit antenna so that coherent addition of the transmit signals is obtained at the receiver side. This provides diversity as well as increases the SNR. The precoder matrix may need to be signaled, by means of feedback signaling and/or signaling of the chosen precoder element in the forward link. The feedback signaling may be viewed as a way for the receiver to provide channel information to the transmitter.
Several different approaches are known for implementing such forward link signaling. For codebook based precoding, explicit signaling of the precoder element index in the forward link is possible. The precoder may also be implicitly signaled using precoded pilots/reference symbols (RS) that together with non-precoded reference symbols may be used at the receiver to determine the used precoder element. Another possibility is to use precoded reference symbols also for the demodulation of the data, that is so-called dedicated RS or alternatively demodulation RS or UE specific RS, and absorb the precoder element into the effective channel from the perspective of the receiver.
As mentioned above, the precoder may be determined/selected with different levels of information of the propagation channel between the transmitter and the receiver. Precoder selection that does not rely on the channel state is often referred to as open-loop precoding and is particularly useful in scenarios where the channel state changes rapidly and is not possible to track with sufficient precision. In more stationary scenarios, closed-loop precoding performs significantly better, because the precoder is adaptively selected to match the state of the channel and thereby maximize the performance.
Closed-loop precoding relies on the availability of channel state information at the transmitter, which must be provided by a feedback mechanism from the receiver. Such feedback may be analogue in the form of sounding signals in the reverse link or digitally signaled over a reverse link. For example, the receiver may select or recommend a precoder (or precoders) from a precoder codebook and feed back the corresponding codebook index to the transmitter, e.g. as in Rel-8 of LTE and which is referred to as implicit feedback in some contexts. A precoder recommendation may be seen as a form of channel quantization since typically a set of channel realizations map to a certain precoding element.
For closed-loop precoding to be effective, the precoder must be well matched to the state of the effective channel, which includes transmit and receive filters, channel responses of antenna cables and the actual propagation channel. This poses a problem in wideband systems where the channel may change over the frequency band (i.e. frequency-selective channels). To match the channel it may be necessary to adaptively change the precoder over the frequency band (frequency-selective precoding), which increases the demand on the frequency resolution of the feedback of channel state information. For example, a separate precoder may have to be recommended for each sub-band, where a sub-band is a frequency segment where a precoder is deemed sufficiently well matched to the channel. Doing so typically results in a significantly larger feedback/signaling overhead.
One particular problem is when the frequency selectivity of the effective channel is much higher than in the underlying radio propagation channels, which could be caused e.g. by: non-calibrated antenna arrays; and distributed antenna systems where the propagation distance from each site to the receiver (or transmitter) is significantly different. In such cases, traditional precoder selection/recommendation should be performed on a significantly higher frequency density than is strictly required by the underlying propagation environment. This is particularly clear in the common case of propagation channels with correlation, for which a single wideband precoder may be sufficient also in frequency selective channels, because the precoder may be tuned to match the statistics of the propagation channel, which may be valid over a significantly wider bandwidth than the coherence bandwidth of the propagation-channel. However, if the antennas and transmit radio chains for example are non-calibrated, the correlation of the propagation channel is not preserved in the effective channel. Instead the statistics of the effective channel change over frequency, requiring frequency selective precoder feedback. Alternatively, the antennas must be calibrated with an associated increase in system cost.
For a system with non-calibrated antennas, let HRP(f) denote the frequency response of the radio-propagation channel. The effective channel may then be modeled as Heff(f)=HRx(f)HRP(f)HTx(f) where HRx(f), where HTx(f) are the frequency responses of the receiver and the transmitter respectively. Generally, the frequency selectivity induced by the receiving antennas and filters, HRx(f), may be accounted for as part of the receive processing because the channel knowledge at the receiver is typically much better than at the transmitter. Moreover HTx(f) typically do not fade over frequency (i.e. the gains do not change) but rather induce phase rotations, which in addition are rather stable over time.
Mismatched transmit antennas and filters are however more problematic because the mismatch causes fast variations in HTx(f), which is problematic for channel dependent closed loop precoding, where the received signal, y(f), may be modeled as y(f)=Heff(f)W(f)x(f) with x(f) being the modulated information carrying symbols. For the precoding to match the effective channel, the frequency-selectivity of the precoder must match the frequency-selectivity of the effective channel.
A common model for the impulse response of the transmitter, which models the transmit delays of each transmit (TX) antenna, is given by HTx(τ)=diag(α1δ(τ−τ1), . . . , αNTxδ(τ−τNTx)) which corresponds to the frequency response
            H      Tx        ⁡          (      f      )        ∝            diag      ⁡              (                                            α              1                        ⁢                          ⅇ                                                -                                      j2πτ                    1                                                  ⁢                f                                              ,          …          ⁢                                          ,                                    α                              N                Tx                                      ⁢                          ⅇ                                                -                                      j2πτ                                          N                      Tx                                                                      ⁢                f                                                    )              .  That is, compared to a calibrated array, having τ1=τ2= . . . =τNTx=0, the effective channel is related as given by:
                                          H            eff                    ⁡                      (            f            )                          =                                            H              eff              calibrated                        ⁡                          (              f              )                                ·                      diag            ⁡                          (                                                ⅇ                                                            -                                              j2πτ                        1                                                              ⁢                    f                                                  ,                …                ⁢                                                                  ,                                  ⅇ                                                            -                                              j2πτ                                                  N                          Tx                                                                                      ⁢                    f                                                              )                                                          (        1        )            As such, the relative phase between the TX antennas is rotated over frequency. For example, the relative phase between antenna m and n is rotated by the phase 2π(τn−τm)f. If the bandwidth B is larger or same order of magnitude as
      1          2      ⁢              πΔτ        max              ,            where      ⁢                          ⁢              Δτ        max              =                  max                  m          ,          n                    ⁢                                            τ            m                    -                      τ            n                                        ,then there is a significant phase rotation within the band. That is, if the maximum tolerated relative phase rotation in a subband is x radians, then the subband bandwidth, BSB, is upper bounded as:
                              B          SB                ≤                  x                      2            ⁢                          πΔτ              max                                                          (        2        )            
Accordingly, for traditional precoding/beamforming, the subband bandwidth in which a precoder is valid is upper bounded in accordance with equation (2). This is in particular restricting for wideband precoding that is matched to the statistics of the channel Reff,Tx(f)=E{HeffH(f)Heff(f)}≈HTxH(f)E{HRPH(f)HRP(f)}HTx(f). It is well known that the statistics of the radio propagation channel are well approximated as constant over the bandwidth RTx,RP=E{HRPH(f)HRP(f)}, and the frequency selectivity of the transmit covariance matrix of the effective channel Reff,Tx(f)=HTxH(f)RTx,RPHTx(f) is thereby more or less completely induced by frequency response of the transmit filters and antennas, HTx(f). In other words, with perfectly calibrated antennas, a precoder/beamformer tuned to the channel statistics is valid over the entire bandwidth, which is highly useful in correlated channel environments. With non-calibrated antennas, the precoder is only valid on subbands of bandwidths limited as given by equation (2).
For a system with multi-site coherent joint transmission, the propagation time difference between the terminal and the different sites may be substantially different. Such propagation time differences may have a severe adverse impact on the subband size in which a precoder is valid, much in the same way that non-calibrated antenna arrays affect the frequency selectivity. Let Hk(τ) be the effective channel impulse response, not including the propagation delay, to the k:th site, and let τk be the propagation delay. The compound channel impulse response, including propagation delays, may then be written as Hcompound(τ)=[H1(τ−τ1) . . . HN(τ−τN)]. The frequency response is readily obtained as:Hcompound(f)=[e−j2πτ1fH1(f), . . . , e−j2πτNfHN(f)]  (3)and it is observed that the subband in which a precoder operating on the compound channel is again limited by equation (2).